3.1.4 Joint Moments

Question 1

What is a joint moment?

Answer 1

A joint moment is the expected value of a function of two random variables, usually written as E[g(X, Y)].

Question 2

How do you calculate a joint moment for discrete variables?

Answer 2

Question 3

What does it mean if g(X, Y) = X?

Answer 3

Then E[g(X, Y)] = E[X], and you can compute it using X’s marginal distribution or via the joint distribution.

Question 4

What does it mean if g(X, Y) = Y?

Answer 4

Then E[g(X, Y)] = E[Y], and you can compute it using Y’s marginal distribution or via the joint.

Question 5

What if g(X, Y) depends on both variables?

Answer 5

You must use the full joint distribution to compute E[g(X, Y)].

Question 6

How can you compute E[X] using the joint distribution?

Answer 6

E[X] = Σ_x x * (Σ_y P(X = x, Y = y)).

Question 7

What’s the advantage of using the joint distribution directly for joint moments?

Answer 7

It avoids needing to compute marginal distributions separately.